Estimating 3-D lighting environments requires, of course, estimation of 3-D surface normals. While this is trivial for simple geometric shapes, it is not always easy or even possible to accurately estimate 3-D geometry from arbitrary objects in a single image. Three-dimensional modeling of faces [1]} and bodies [2]}, however, has become increasingly more reliable and accurate, allowing for the recovery of 3-D lighting environments from people in images. When 3-D models cannot be built, a 2-D lighting environment [3]} (or direction [4]}, [5]}) can be estimated by observing that five of the nine spherical harmonics – \(Y_{0,0}(\cdot )\) , \(Y_{1,-1}(\cdot )\) , \(Y_{1,1}(\cdot )\) , \(Y_{2,-2}(\cdot )\) , and \(Y_{2,2}(\cdot )\) , Equation (REF ) – do not depend on the \(z\) -component of the required 3-D surface normal. As such, a 2-D, reduced dimensional, lighting environment can be estimated from only 2-D surface normals easily extracted from the boundary of arbitrary objects (where the \(z\) -component of the surface normal is 0).
